The integrality conjecture and the cohomology of preprojective stacks

We study the Borel-Moore homology of stacks of representations of preprojective algebras $Π_Q$, via the study of the DT theory of the undeformed 3-Calabi-Yau completion $Π_Q[x]$. Via a result on the supports of the BPS sheaves for $Π_Q[x]$-mod, we prove purity of the BPS cohomology for the stack of $Π_Q[x]$-modules, and define BPS sheaves for stacks of $Π_Q$-modules. These are mixed Hodge modules on the coarse moduli space of $Π_Q$-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure, and thus that the Borel-Moore homology of stacks of $Π_Q$-modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of $Π_Q$-modules. Among these and other applications, we use our results to prove positivity of a number of "restricted" Kac polynomials, determine the critical cohomology of $\mathrm{Hilb}_n(\mathbb{A}^3)$, and the Borel-Moore homology of genus one character stacks, as well as various applications to the cohomological Hall algebras associated to Borel-Moore homology of stacks of preprojective algebras, including the PBW theorem, and torsion-freeness.

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